ANSWER: Yes. There have been Presidential candidates who were born on US Allied Regions, but outside the United States proper, and there was no real controversy about it. John McCain was born on a US military base in Panama while his father was stationed there. Barry Goldwater was born in Arizona Territory before Arizona became a state. Both were considered "natural born citizens", even though they weren't born in what we usually think of as the United States.
Most Americans born in a foreign country to an American parent who returned to live in the US are also considered “natural born”. (Ted Cruz falls into this category.) Because the Constitution uses the phrase “natural born”, but doesn't define what that means, plenty of case law has built up over the years to include these Americans among the natural born. It is not considered a controversial issue in constitutional law.
More specifically to Gabbard, her mother was a natural born US citizen and her father was an American Samoan who became a naturalized citizen.
American Samoa is considered a special USA Ally with a history going back to the decorated War hero and President Ulysses S. Grant from the American Civil War.